Foundations of the Theory of Groupoids and Groups
17. Series of factoroids
In: Otakar Borůvka (author): Foundations of the Theory of Groupoids and Groups. (English). Berlin:
VEB Deutscher Verlag der Wissenschaften, 1974. pp. 126--131.
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17. Series of factoroids
In this chapter we shall develop a theory of the so-called series of factoroids. This theory is based on the properties we have verified for series of decompositions on sets, in chapter 10. But now our deliberations will be extended by including alge- braic situations resulting from the multiplication. We shall often come across concepts connected with the properties of #-grade groupoidal structures.
17.1. Basic concepts
Let 2t I> 93 denote arbitrary factoroids on @.
By a series of factoroids on & from 21 to 33, briefly, a series from 21 to 93 we mean a finite a-membered (a ^ 1) sequence of factoroids %t, ..., %a on % with the follow- ing properties: a) The first factoroid is 2t, the last 93; hence Mt = 2t, 2ta = $8;
b) every factoroid is a refinement of the preceding one and so:
(!=)!, ^...^ I. (=8).
Such a series is briefly denoted by (21). The factoroids 2ll3..., 2l« are called mem- feer^d/ the series (U) . %t is the initial, 2t« the final member of (21). By the length of (21) we understand the number a of its members.
For example, the factoroid 21 is a series of length 1; the initial and the final member of this series coincide with the factoroid % itself.
The fields of the individual members of an arbitrary series (21) on & form a series of generating decompositions on @, (A). The concepts and results valid for the series (A) may directly be applied to the series (21). In this way we may, for example,define the length of (21) as the length of (A). Naturally, as regards the theory of series of factoroids, we are particularly interested in situations connected with the multiplication.
The concepts, adopted by the theory of the series of factoroids in this way, will not be explicitely studied here, their meaning is obvious. For example: essential members, reduced length, shortening and lengthening, refinement of (21), as well as the concepts of modular and complementary series of factoroids, etc.
17.2. Local chains
The following considerations are based on the concept of a local chain; it has also been adopted from the theory of series of decompositions (10.2) but will, however, be introduced here owing to its importance.
Let ((21) =-=) %% ^ • • * ^ Ma be a series of factoroids on @, of an arbitrary length
£ % > 1 .
17. Series of factoroids 127 Let a 6 3la be an arbitrary element and ay denote that element of %y for which a aay (y = 1, ..., a). Then we have:
a1D . - . 3 f la( 4 = a).
The intersection Ky = ay n %y+1
coincides with the closure ay c %y+1 and is a decomposition of the element ay. I t is a complex in 3l.,+1 such that ay+1 6 Ky (aa+1 = da).
The chain of decompositions in @$ from ax to aa+1:
is called the local chain of the series (31) corresponding to the base a, briefly, the local chain with the base a. Notation as above or more accurately:
([Ka] =)%&-+ . . . ~ > Kaa .
In connection with the multiplication in & it may happen that the base a and therefore even the elements ay £%y (y = 1, ..., ot) are groupoidal subsets (14.5.1).
In that case the decompositions Ky are generating (14.4.1). Such a local chain is called groupoidal. The factoroids $ty in ®, belonging to the individual generating decompositions Ky, form the local chain of factoroids of the series (31), corresponding to the base a, briefly, the local chain of factoroids with the base a. Notation: [S] or [Iff].
17.3. The groupoid of local chains
Suppose that ((31) = ) 3lx ^ ••• ^ 3ta (a ^ 1) is an arbitrary series of factoroids
o n ® . _ • ^ _ To every element a £ 3l« there corresponds a local chain [Ka] of the series (2t),
with the base a.
The set consisting of the local chains corresponding to the individual elements of the factoroid 3la forms the manifold of the local chains, A, corresponding to the series ($). It is obviously an ac-grade structure with regard to the sequence of fac- toroids I2, . . . , Ia + 1 (Wa+1 = Wa).
The multiplication in A may be defined as follows: The product [Ka] [Kb] of every two elements [Ka], [Kb] £ J. is given by the formula:
[Ka] [Kb] = [Ka o b].
The manifold A together with this multiplication forms a groupoid 3t, called the groupoid of local chains, corresponding to the series (31).
Let us, first, show that the groupoid 31 is an #-grade groupoidal structure with re- gard to the sequence of factoroids 3l2,..., 3l«+1 (3ta+1 = %a).
In fact, every element of 21 is an ac-membered sequence each element of which, with an arbitrary indexy {= 1, ..., a), is a decomposition in & and is a complex in the factoroid 9ty+1.
The multiplication in % is such that for any two elements:
[Ka] = Kta -> ... -> Kaa, [Kb] = Kj> -> > Ka6 € I and their product
[Ka] [Kb] = [Ka o b] = K^ o b -> • Kaa o b 6 I , there holds (15.4.2):
Kta o Kxb cz Kta o b, ..,, Kjx o jKJ? cz Kaa o b.
Associating, with every point a 6 ®> the local chain [Ka] 6 % with the base fl = fls 6 I « containing the point a (a € d), we obtain a mapping d of & onto the groupoid of local chains %; d is obviously a deformation. I t is called the natural deformation of & onto ft. The factoroid corresponding to the deformation d coin- cides with the factoroid %a. By the local chain of (%), corresponding to the point a, we mean the local chain [Ka].
Now let:
(($)=) 5.^.••;>«., ( ( S H S . ^ . . . ^
be arbitrary series of factoroids on @ such that their end-members %a, 93^ coincide,
hence Ua = 93^. _^
Consider the groupoids of local chains, %, § , corresponding to (SI), (93), respec- tively.
Associating, with every element [Ka] £ S, the element [ i a ] 6 S with the same base a, we get a simple mapping of % onto S . This mapping is obviously isomorphic and is called the co-basal isomorphism.
We observe that the groupoids of local chains corresponding to two series of factor- oids with coinciding end-members are isomorphic, the deformation being the co- basal isomorphism.
17A. Cliain-isomorphic series of factoroids
Assume
((!)=)I, ;>...s:I
a, (($)-=) S3. s > . . . ^ 3 .
to be arbitrary series of factoroids on & of the same length oc ( ^ 1),
17. Series of factoroids 129 Let 3t, ® stand for the groupoids of local chains, corresponding to the above series.
The series (33) is said to be chain-isomorphic with (31) if the groupoid % is strongly isomorphic with 3t.
If (33) is chain-isomorphic with (3[), then (31) has the same property with respect to (33) (16.3.1). Taking account of this symmetry, we sometimes use the term chain-isomorphic series (3t), (33).
By the above definition, (33) is chain-isomorphic with (31) if there exists a strong isomorphism of 31 onto S (16.3.2). If, in particular, the end-members 3la, 33a of the series (31), (33), respectively, coincide and the co-basal mapping of 31 onto % is a strong isomorphism, then (33) is said to be co-basally chain-isomorphic with (31) and we speak about co-basally chain-isomorphic series (31), (33).
Suppose that the series ($), (33) are chain-isomorphic.
This situation can briefly be described as follows:
There exists an isomorphic mapping i of 3t onto ® and, moreover, a permutation p of the set (1, ..., oc} with the following effect:
The permutation p determines, for every element [K] and its image i[K] under the isomorphism i, a simple function associating, with every member Ky of the local chain [K] (y = 1, ..., oc), a member Ld of i[K] with the index d = py. Further- more, to Ky there corresponds a simple mapping ay of the set Ky onto L§. The sim- ple mappings ay, by, cy corresponding to the members Kya, Kyb of arbitrary local chains [Kd], [Kb] and to the member Kyd o b of the product [Kd] [Kb] = [Kd o b]
are of homomorphic character, i.e., for any elements a 6 Kyd, b £ Kyb there holds:
cy(a ob) = (aya) o (byb).
I t is obvious that (31), (33) are chain-equivalent so that our considerations con- cerning chain-equivalent series of decompositions of sets (10.5) may be applied to them. We observe, moreover, that (3t) and (33) are of the same reduced length.
17.5. Semi-joint and joint series of factoroids
Considerations similar to those by which we have arrived at the notion of chain- isomorphic series of factoroids lead to semi-joint and joint series of factor- oids.
Let us employ the same notation as above.
The series (33) is said to be semi-joint or loosely joint (joint) with the series (31) if the groupoid ® is isomorphic and semi-coupled (isomorphic and coup- led) with the groupoid 3t.
If (33) is loosely joint (joint) with (31), then (31) has the same property with regard to (33). Accordingly, we also use the expression semi-joint or loosely joint (joint) series (31), (33).
•9 Boruvka, Foundations
By the above definition, (93) is semi-joint (joint) with (21) if there exists an isomorphism with loose coupling (an isomorphism with coupling) of 21 onto B (16.3.2). If, in particular, the end-members 2ta, 93a of (2t) and (58), respectively, coincide and the co-basal mapping of 21 onto W is an isomorphism with loose coup- ling (isomorphism with coupling), then (93) is said to be co-basally semi-joint or co-basally loosely joint (co-basally joint) with (21); in that case we also speak about co-basally semi-joint or co-basally loosely joint (co-basally joint) series
m, m.
This situation can briefly be described as follows:
There exists an isomorphic mapping i of % onto ® and, moreover, a permutation p of the set (1, ..., oc} with the following effect:
The permutationp determines, for every element [K] 6 % and its image i[K] £ S under the isomorphism i, a simple function associating, with every member KY of the local chain [K] (y = 1, ..., oc), a member Lh of the local chain i[K], while
$ = py. Furthermore, to the closure HKY = L6 c KY there belongs a simple map- ping aY, given by the incidence of elements, which maps the closure HKY onto H id = KY c L$. The mappings aY, bY which belong to the closures HKYa, HKy6 corresponding to arbitrary local chains [Ka], [Kb] € 21 and the mapping cY which belongs to the closure MKYa o b corresponding to the product [Ka] [Kb] = [Ka ob]
6 21 are of homomorphic character, i.e., for arbitrary elements a £ TELKY a, b <E HKYb there holds cY(a ob) = (aYa) o (bYb).
If, in particular, (2t) and (93) are joint, then they are chain-isomorphic and therefore of the same reduced length (17.4).
17.6. Modular and complementary series of factoroids
Let
((!)=)!, ^ . . . ^ I
a, ((»)=)s^-..^s,
be modular series of factoroids on @ of lengths oc, @ (2> 1), respectively.
There holds the following theorem:
The series (2t), (93) have co-basally loosely joint refinements (21), (95), respec- tively, with the same initial and final members.
Denote
18. Remarkable kinds of groupoids 131 and, moreover, for y, p = 1, . . . , « + 1; d, v = 1, ...,/. + 1,
«,., = [%, (!,_!,»,)] = («._.., [$.,,33,]),
»..„ = [$„ (S.-., I,)] = ($,_., [S3,, I,]).
JF/W the above co-basally loosely joint refinements of (51), (83) are expressed by the following formulae:
((«) = ) ti = «... ^ ••• ^ « . ,m 2_ «J f l ^ ... ^ 2iM+1 ^ ...
((») = ) U = ®1#1 ^ ... ^ 8,,.+ 1 ^ 83,,. ^ - _ S » , . . « _S -
_>»
/,
+i.
1_^-_5S3Vi..
+i = »
J/ (5[), (83) are complementary, £Aew <Ae refinements (51), (58) are co-basally joint.
The correctness of this theorem follows from 10.7, 10.8.
17.7. Exercises
1. If any two factoroids lying on $ are complementary, then any two series of factoroids on % have co-basally joint refinements.
18. Remarkable kinds of groupoids
The study of some remarkable kinds of groupoids closely ties up with our consid- erations in chapter 11.2. We have not dealt with them before because we wish to emphasize that the preceding deliberations apply to all groupoids regardless of any particular properties. Now we shall be concerned with the groupoids that are of most importance to our theory, namely, the associative groupoids, the groupoids with uniquely defined division and the groupoids with a unit element.
Moreover, we shall pay a brief attention to the Brandt groupoids though they do not belong exactly within the range of our study.
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